“Almost Perfect” Equations of State and Adiabatic Exponents

Friday, Jan. 31, 2025

astrophysics of stars and planets - spring 2025 - university of arizona, steward observatory

Today’s Agenda

1. Announcements - Grades,Volunteers for Solutions, Uploading HW2 (2m)

2. Reading Overview/Key Points (10m)

3. In-Class HW1 Review (Groups of ~4) (20m)

4. In-Class HW1 Presentations (Groups of ~4) (15m)

5. Debrief/Reminders (2m)

“Almost Perfect” Equations of State

In real gases, interactions have to be taken into account that modify the “perfect” results given above. In addition, a stellar equation of state might consist of many components with radiation, Maxwell–Boltzmann, and degenerate gases competing in importance.

A measure of the interaction energy between two ions is the Coulomb potential

Definition 29

Coulomb potential between two ions

\[ \phi = \frac{Z^{2}e^{2}}{a} \]

where \(a\) is the typical seperation between the two ions. Similar to past estiamtes, we suggest that these effects become important when they are comparable to the thermal energy \(kT\). This allows us to define the ratio,

Definition 30

\[ \Gamma_{c} = \frac{Z^{2}e^{2}}{akT} \]

Plotting the different regimes we find:

A composite showing how the ρ–T plane is broken up into regions dom- inated by pressure ionization, degeneracy, radiation, ideal gas, crystallization, and ionization-recombination. The gas is assumed to be pure hydrogen. From HKT 3.9.

Observation 4

The axis are flipped, \(\rm{log}T\) on the x-axis in this example!

Roughly, for \(\Gamma_{v}\gt1\) Coulumb effects may be important. For values of \(\Gamma_{v}\gt170\) and appropriate densities such as in white dwarfs, Coulomb effects overwhelm those of thermal agitation and the gas settles down into a crystal. Also noted is the boundary at which the Saha equation assumptions terminate - pressure ionization.

Adiabatic Exponents and Other Derivatives

3.7.1 Keeping the Composition Fixed

Specific Heats

Definition 31

general form of specific heat

\[ c_{\alpha} = \left ( \frac{dQ}{dT} \right )_{\alpha} \]

where \(\alpha\) is kept fixed as \(T\) changes.

Units

\(Q\) - (\(\rm{erg \ g}^{-1}\))

\(c_{\alpha}\) - (\(\rm{erg \ g^{-1} K^{-1}}\))

From HKT Eq. 1. we have:

Definition 32

\[ dQ = dE + P dV_{\rho} = dE + Pd(\frac{1}{\rho}) = dE - \frac{P}{\rho^2} d\rho \]

where \(V_{\rho} = 1/\rho\) is the specific volume.

This leads us to

Definition 33

the specific heat at constant-volume

\[ c_{V_{\rho}} = \left ( \frac{dQ}{dT} \right )_{\rho} = \left ( \frac{\partial E}{ \partial T} \right )_{\rho} (\rm{erg \ g^{-1} K^{-1}}) \]

for an ideal monotomic gas we have \(E=3N_{\rm{A}}kT/2\mu\) such that \(c_{V_{\rho}}=3N_{\rm{A}}k/2\mu\) or \(E=c_{V_{\rho}} T\).

To find the specific heat at constant pressure we start with the relation:

Definition 34

Relation between specific heats

\[ c_{P} - c_{V_{\rho}} = -T \left ( \frac{\partial P}{ \partial T} \right )^{2}_{\rho \rm{or} V_{\rho}} \left ( \frac{\partial P}{ \partial V_{\rho}} \right )^{-1}_{T} \]

using the power-law expression for pressure from HKT 1.67, \(P = P_{0}\rho^{\chi_{\rho}}T^{\chi_{T}}\).

This gives the following definitions for the constants \(\chi_{T}\) and \(\chi_{\rho}\):

Definition 35

\[ \chi_{T} = \left ( \frac{\partial \rm{ln} P}{ \partial \rm{ln} T} \right )_{\rho \rm{or} V_{\rho}} = \frac{T}{P} \left ( \frac{\partial P}{ \partial T} \right )_{\rho \rm{or} V_{\rho}} \]

and

Definition 36

\[ \chi_{\rho} = \left ( \frac{\partial \rm{ln} P}{ \partial \rm{ln} \rho} \right )_{T} = - \left ( \frac{\partial \rm{ln} P}{ \partial \rm{ln} V_{\rho}} \right )_{T} = \frac{\rho}{P} \left ( \frac{\partial P}{\partial \rho} \right )_{T} = - \frac{1}{\rho P} \left ( \frac{\partial P}{\partial V_{\rho}} \right )_{T} \]

Finally, combining this all together we find a reduced expression:

Definition 37

Simplified relation between specific heats

\[ c_{P} - c_{V_{\rho}} = \frac{P}{\rho T} \frac{\chi^2_{T}}{\chi_{\rho}} \]

For an Ideal Gas, \(\chi_{T}=\chi_{\rho}=1\) giving

\[ c_{P} - c_{V_{\rho}} = \frac{N_{\rm{A}} k}{\mu} \]

and finally \(c_{P}=5N_{\rm{A}}k/2\mu\).

We also define a new variable

Definition 38

Ratio of specific heats

\[ \gamma = \frac{c_{P}}{c_{V_{\rho}}} \equiv 1 + \frac{P}{\rho T c_{V_{\rho}}} \frac{\chi^2_{T}}{\chi_{\rho}} \]

Adiabatic Exponents

The dimensionless adiabatic exponents, the \(\Gamma\)’s measure the thermodynamic response of the system to adiabatic changes and will be used extensively. As in Chapter 1, the subscript “ad” means that the indicated partials are to be evaluated at constant entropy.

Definition 39

\[ \Gamma_{1} = \left ( \frac{\partial \rm{ln} P}{\partial \rm{ln} \rho} \right )_{\rm{ad}} = - \left ( \frac{\partial \rm{ln} P}{\partial \rm{ln} V_{\rho}} \right )_{\rm{ad}} \]

Definition 40

\[ \frac{\Gamma_{2}}{\Gamma_{2}-1} = \left ( \frac{\partial \rm{ln} P}{\partial \rm{ln} T} \right )_{\rm{ad}} = \frac{1}{\nabla_{\rm{ad}}} \]

Definition 41

\[ \Gamma_{3}-1 = \left ( \frac{\partial \rm{ln} T}{\partial \rm{ln} \rho} \right )_{\rm{ad}} = - \left ( \frac{\partial \rm{ln} T}{\partial \rm{ln} V_{\rho}} \right )_{\rm{ad}} \]

Finally, we have

\[ \frac{\Gamma_{3} - 1}{\Gamma_{1}} = \frac{\Gamma_{2} - 1}{\Gamma_{2}} = \nabla_{\rm{ad}}. \]

Now, we can write some more useful versions of these exponents and relating to the specific heats:

Definition 42

\[ \Gamma_{3}-1 = \frac{P}{\rho T} \frac{\chi_{T}}{c_{V_{\rho}}} = \frac{1}{\rho} \left ( \frac{\partial P}{\partial E} \right )_{\rho} \]

Definition 43

\[ \Gamma_{1} = \chi_{T} (\Gamma_{3}-1) + \chi_{\rho} = \frac{\chi_{\rho}}{1-\chi_{T} \nabla_{\rm{ad}}} \]

Definition 44

\[ \frac{\Gamma_{2}}{\Gamma_{2}-1} = \nabla^{-1}_{\rm{ad}} = c_{P} \frac{\rho T}{P}\frac{\chi_{\rho}}{\chi_{T}} = \frac{\chi_{\rho}}{\Gamma_{3}-1} + \chi_{T} \]

and lastly, \(\gamma\)

Definition 45

\[ \gamma = \frac{c_{P}}{c_{V_{\rho}}} = \frac{\Gamma_{1}}{\chi_{\rho}} = 1 + \frac{\chi_{T}}{\chi_{\rho}}(\Gamma_{3}-1) = \frac{\Gamma_{3}-1}{\chi_{\rho}} \frac{1}{\nabla_{ad}} \]

Mixtures of Ideal Gases and Radiation

For stellar models, an equation of state that is a mixture of ideal gas and radiation can suffice, allowing us to write the pressure as:

Definition 46

\[ P_{\rm{total}} = \frac{\rho N_{\rm{A} kT}}{\mu} + \frac{a T^4}{3} = P_{g} + P_{rad}. \]

and the energy in a similar fashion

Definition 47

\[ E_{\rm{total}} = \frac{3 N_{\rm{A} kT}}{2\mu} + \frac{a T^4}{\rho} = P_{g} + P_{rad}. \]

Observation 5

Be sure to keep track of \(V\) and \(V_{\rho}\) when using any of the above equations!

In-Class Activity

1. Form groups of 3-4

2. Choose a problem from the HW1 (Exercise 1 or 2).

3. Compare solutions, nominate a scribe that will write up the solutions and that you all agree on the answer!

4. This person will have 5 minutes at the end class to share a condensed version of the solution for the class.

Learning Objective

• discuss exercises with others to see if you agree with the logic / steps OR identify where your steps diverge